6.4 Rhombuses, Rectangles, and Squares
Warm up: ABCD is a parallelogram. Find each measure. A. CD B. mC 104°
Objective Use properties of special types of parallelograms.
Rhombus A rhombus is a with four congruent sides. The diagonals are perpendicular The diagonals bisect opposite angles Looks like a parallelogram but has more
Rhombus
Rectangle A rectangle is a with four right angles The diagonals of a rectangle are congruent The adjacent sides are perpendicular (right angle)
Square A square is a with four congruent sides AND four right angles The diagonals of a square are congruent The diagonals are perpendicular (right angles)
1) In the diagram, ABCD is a rectangle. a. Find AD and AB. b. Find m∠A, m∠B, m∠C, and m∠D. 8 5 All angles are 90 degrees.
2) In the diagram, PQRS is a rhombus. Find QR, RS, and SP. 6 6 6
Corollaries A corollary- to a theorem is a statement that can be proved easily using the theorem.
Rhombus corollary If a quadrilateral has four congruent sides, then it is a rhombus.
Rectangle Corollary If a quadrilateral has four right angles, then it is a rectangle.
Square Corollary If a quadrilateral has four congruent sides AND right angles, then it is a square.
Sides are not congruent Use the information in the diagram to name the special quadrilateral. Four right angles Sides are not congruent Must be a rectangle
Sides are all congruent 2. Use the information in the diagram to name the special quadrilateral. No info on angles Sides are all congruent Must be a rhombus
Four right angles Four congruent sides Must be a square 3. Use the information in the diagram to name the special quadrilateral. Four right angles Four congruent sides Must be a square
Remember: We wrote this before but it is important to remember that…. …the diagonals of a rhombus are perpendicular
ABCD is a rhombus. Find the value of x. 1) We know that the diagonals are perpendicular, so lets add a box by E.
ABCD is a rhombus. Find the value of x. 2) We know that the diagonals are perpendicular, so lets add a box in the center . X = 90 degrees
Remember: We wrote this before but it is important to remember that…. …the diagonals of a rectangle are congruent
Opposite sides are congruent 1a) You nail four pieces of wood together to build a four-sided frame, as shown. What is the shape of the frame? No info on angles Opposite sides are congruent Must be a parallelogram
1b) The diagonals measure 7 ft 4 in. and 7 ft 2 in 1b) The diagonals measure 7 ft 4 in. and 7 ft 2 in. Is the frame a rectangle? PYTHAGOREAN THEOREM Notice the triangle is a right triangle What formula do we use to solve for missing sides of a RIGHT triangle?
2) rectangle EFGH . Find the value of x. The pieces of the diagonals of a rectangle are all all congruent x=12
3) square JKLM. Find the value of x. The diagonals of a square are perpendicular